ISMAGS 算法#

提供了 ISMAGS 算法的 Python 实现。[1]

它能够找到两个图之间的（子图）同构，同时考虑子图的对称性。在大多数情况下，VF2 算法比此实现更快（至少在小图上），但在某些情况下存在指数级的对称等价同构。在这种情况下，ISMAGS 算法将为每个对称群仅提供一个解。

>>> petersen = nx.petersen_graph()
>>> ismags = nx.isomorphism.ISMAGS(petersen, petersen)
>>> isomorphisms = list(ismags.isomorphisms_iter(symmetry=False))
>>> len(isomorphisms)
120
>>> isomorphisms = list(ismags.isomorphisms_iter(symmetry=True))
>>> answer = [{0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5, 6: 6, 7: 7, 8: 8, 9: 9}]
>>> answer == isomorphisms
True

此外，此实现还提供了查找任意两个图之间最大公共导出子图 [2] 的接口，同样考虑了对称性。给定图 graph 和 subgraph，算法将从 subgraph 中移除节点，直到 subgraph 与 graph 的子图同构。由于只考虑了 subgraph 的对称性，因此值得思考如何提供图。

>>> graph1 = nx.path_graph(4)
>>> graph2 = nx.star_graph(3)
>>> ismags = nx.isomorphism.ISMAGS(graph1, graph2)
>>> ismags.is_isomorphic()
False
>>> largest_common_subgraph = list(ismags.largest_common_subgraph())
>>> answer = [{1: 0, 0: 1, 2: 2}, {2: 0, 1: 1, 3: 2}]
>>> answer == largest_common_subgraph
True
>>> ismags2 = nx.isomorphism.ISMAGS(graph2, graph1)
>>> largest_common_subgraph = list(ismags2.largest_common_subgraph())
>>> answer = [
...     {1: 0, 0: 1, 2: 2},
...     {1: 0, 0: 1, 3: 2},
...     {2: 0, 0: 1, 1: 2},
...     {2: 0, 0: 1, 3: 2},
...     {3: 0, 0: 1, 1: 2},
...     {3: 0, 0: 1, 2: 2},
... ]
>>> answer == largest_common_subgraph
True

然而，如果不考虑对称性，则无关紧要

>>> largest_common_subgraph = list(ismags.largest_common_subgraph(symmetry=False))
>>> answer = [
...     {1: 0, 0: 1, 2: 2},
...     {1: 0, 2: 1, 0: 2},
...     {2: 0, 1: 1, 3: 2},
...     {2: 0, 3: 1, 1: 2},
...     {1: 0, 0: 1, 2: 3},
...     {1: 0, 2: 1, 0: 3},
...     {2: 0, 1: 1, 3: 3},
...     {2: 0, 3: 1, 1: 3},
...     {1: 0, 0: 2, 2: 3},
...     {1: 0, 2: 2, 0: 3},
...     {2: 0, 1: 2, 3: 3},
...     {2: 0, 3: 2, 1: 3},
... ]
>>> answer == largest_common_subgraph
True
>>> largest_common_subgraph = list(ismags2.largest_common_subgraph(symmetry=False))
>>> answer = [
...     {1: 0, 0: 1, 2: 2},
...     {1: 0, 0: 1, 3: 2},
...     {2: 0, 0: 1, 1: 2},
...     {2: 0, 0: 1, 3: 2},
...     {3: 0, 0: 1, 1: 2},
...     {3: 0, 0: 1, 2: 2},
...     {1: 1, 0: 2, 2: 3},
...     {1: 1, 0: 2, 3: 3},
...     {2: 1, 0: 2, 1: 3},
...     {2: 1, 0: 2, 3: 3},
...     {3: 1, 0: 2, 1: 3},
...     {3: 1, 0: 2, 2: 3},
... ]
>>> answer == largest_common_subgraph
True

注意#

  • 当前实现仅适用于无向图。但该算法通常也应适用于有向图。

  • 提供的两个图的节点键都需要完全可排序和可哈希。

  • 假定节点和边的相等性是可传递的：如果 A 等于 B，且 B 等于 C，则 A 等于 C。

参考文献#

[1]

M. Houbraken, S. Demeyer, T. Michoel, P. Audenaert, D. Colle, M. Pickavet, “The Index-Based Subgraph Matching Algorithm with General Symmetries (ISMAGS): Exploiting Symmetry for Faster Subgraph Enumeration”, PLoS One 9(5): e97896, 2014. https://doi.org/10.1371/journal.pone.0097896

ISMAGS 对象#

ISMAGS(graph, subgraph[, node_match, ...])

实现了 ISMAGS 子图匹配算法。
在本页上